HAL Id: hal-00980984
https://hal.archives-ouvertes.fr/hal-00980984v2
Submitted on 15 Sep 2014
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Characterization of function spaces via low regularity
mollifiers
Xavier Lamy, Petru Mironescu
To cite this version:
REGULARITY MOLLIFIERS
Xavier Lamy
Universit´e de Lyon, CNRS UMR 5208, Universit´e Lyon 1, Institut Camille Jordan 43, blvd. du 11 novembre 1918
F-69622 Villeurbanne cedex, France
Petru Mironescu
Universit´e de Lyon, CNRS UMR 5208, Universit´e Lyon 1, Institut Camille Jordan 43, blvd. du 11 novembre 1918
F-69622 Villeurbanne cedex, France
Abstract. Smoothness of a function f : Rn→ R can be measured in
terms of the rate of convergence of f ∗ ρεto f , where ρ is an appropriate
mollifier. In the framework of fractional Sobolev spaces, we characterize the “appropriate” mollifiers. We also obtain sufficient conditions, close to being necessary, which ensure that ρ is adapted to a given scale of spaces. Finally, we examine in detail the case where ρ is a characteristic function.
1. Introduction
The smoothness of a function f : Rn → R can be measured by different decay properties, for example via the decay properties of its harmonic ex-tension, or the ones of its Littlewood-Paley decomposition, or the ones of its coefficients in an appropriate wavelets frame. See [7, Chapter 2] for a thorough discussion on this subject. Another characterization is related to the rate of convergence of f ∗ ρε to f , where ρ is an appropriate mollifier.
For example, for non integer s > 0 and 1 ≤ p < ∞ we have (1) kf kpWs,p ∼ kf k p Lp+ ˆ 1 0 1 εsp+1kf −f ∗ρεk p Lpdε, where ρε(x) = 1 εnρ x ε , provided (2) ρ ∈ S and ˆ ρ = 1.
Here S denotes the Schwartz class of smooth, rapidly decreasing functions. We address here the question of the validity of 1 under assumptions as weak as possible on ρ. This is a “continuous” (vs “discrete”) counterpart of
1991 Mathematics Subject Classification. Primary: 46E35.
Key words and phrases. Besov spaces, approximation, mollifiers, Littlewood-Paley decomposition.
the analysis of Bourdaud [1] concerning the minimal assumptions required on the (father and mother) wavelets appropriate for the characterization of Besov spaces.
Usually, the assumption ρ ∈ S is weakened as follows. First, validity of
1 is established for some eρ ∈ S. Next, one expresses an arbitrary ρ in the form
(3) ρ =X
j≥0
ηj∗ eρ2−j [4, Lemma 2, p. 93].
Then, using 3and the validity of2 for eρ, it follows that property1holds for ρ provided the ηj’s decay sufficiently fast. Finally, decay of ηj is obtained by requiring a sufficient decay of the Fourier transform bρ of ρ. With more work, spatial conditions on ρ (of Fourier multiplier’s theorem type) ensure the decay of bρ and thus lead to (usually suboptimal) sufficient conditions for the validity of1.1 Alternatively, in standard function spaces one can rely on the decomposition of functions in simple building blocks (e.g. atoms) and obtain almost sharp spatial sufficient conditions. For such an approach in the framework of the Hardy spaces, see [5], [3].
In what follows, we will obtain, using very little technology, necessary and sufficient conditions on ρ in order to have 1, and simple sufficient spatial conditions on ρ, close to being optimal.
Of special interest to us will be the validity of1when f ∗ ρεis particularly
simple to compute. A typical example consists in taking ρ the characteristic function of a unit cube, e.g. Q = (0, 1)n or Q = (−1/2, 1/2)n. We will determine the spaces Ws,p which can be described via such a ρ.
It turns out that our techniques are adapted not only to the Sobolev spaces with non integer s, but more generally to the Besov spaces Bsp,qwith s > 0, 1 ≤ p ≤ ∞ and 1 ≤ q ≤ ∞. Recall that this scale of spaces includes the one of fractional Sobolev spaces, since Ws,p= Bp,ps for non integer s [7, Chapter 2]. For simplicity, we will write all our formulas and statements only when q < ∞. However, our results hold also when q = ∞, and the corresponding results are obtained by straightforward adaptations of the formulas and arguments.
Our first result is a one sided estimate, which surprisingly requires no smoothness of ρ.
Theorem 1.1. Let ρ ∈ L1 be such that ´ ρ = 1. Then for every s > 0, 1 ≤ p ≤ ∞ and 1 ≤ q < ∞ we have (4) kf kqBs p,q .kf k q Lp+ ˆ 1 0 1 εsq+1kf − f ∗ ρεk q Lpdε.
Remark 1. It is tempting to extend Theorem 1.1 to finite measures, but the example ρ = δ (the Dirac mass at the origin) shows that Theorem 1.1
1A typical result for which this approach is followed is the fact that the norm on the
Besov spaces Bs
p,q does not depend on the choice of the rapidly decreasing mollifier; see
need not hold for a measure. We do not know how to characterize the finite measures of total measure 1 satisfying 4.
We next discuss what is needed in order to obtain the reverse of 4. For this purpose, we fix some η ∈ S. Assuming that the reverse of 4 holds, we have (5) ˆ 1 0 1 εsq+1kη − η ∗ ρεk q Lpdε < ∞.
It turns out that5 with p = q = 1 is also sufficient.
Theorem 1.2. Let ρ ∈ L1 satisfy ´ ρ = 1. Let s > 0. Then the following
are equivalent.
(1) There exists some η ∈ S such that ´ η 6= 0 and (6)
ˆ 1
0
1
εs+1kη − η ∗ ρεkL1dε < ∞.
(2) For every 1 ≤ p ≤ ∞ and every 1 ≤ q < ∞ we have
(7) kf kqBs p,q ∼ kf k q Lp+ ˆ 1 0 1 εsq+1kf − f ∗ ρεk q Lpdε.
An additional equivalent characterization of ρ satisfying the above prop-erties will be provided in Section 4.
We now turn to the case where ρ is the characteristic function of a set A. In that case, the range of values of s for which the equivalent characterizations of Theorem 1.2 are satisfied depends only on whether or not the set A is centered:
Proposition 1. Let ρ = 1
|A|1A, where A ⊂ R
n is a bounded measurable set
of positive Lebesgue measure. Then ρ characterizes all the spaces Bp,qs for fixed s (that is, 7 is valid) if and only if:
(1) Either´
Ay dy = 0 and s < 2.
(2) Or´
Ay dy 6= 0 and s < 1.
Finally, we provide sufficient spatial conditions for the validity of7 when 0 < s < 1.
Proposition 2. Let ρ ∈ L1 satisfy´ ρ = 1, and 0 < s < 1. If ρ satisfies the
moment condition
(8) ˆ
|y|s|ρ(y)| dy < ∞,
then ρ characterizes all spaces Bp,qs . That is, 7 is valid.
For s ≥ 1, the exemple of ρ = 1A with uncentered A shows that there
is no such simple sufficient finite moment condition. In order to obtain the validity of7for higher s, one would need to ask for the vanishing of moments, as in the case of ρ = 1A. For more details see Proposition4 below.
Proposition 3. Let s > 0. Let ρ ∈ L1 satisfy ´ ρ = 1 and ρ ≥ 0. If 7 is
valid, then ρ necessarily satisfies the moment condition 8.
The plan of the paper is as follows. In Section 2 we introduce some pre-liminary notation, definitions and tools required in the sequel. In Sections3
and 4 we prove our two main results, Theorems 1.1 and 1.2. Eventually, Section 5is devoted to proving Propositions1,2 and 3.
2. Preliminaries 2.1. Littlewood-Paley decomposition and Bs
p,q. We will make use of
the (inhomogeneous) Littlewood-Paley decomposition of a temperate distri-bution. Let ζ, ϕ ∈ S(Rn) be as follows:
• supp bζ ⊂ B(0, 2) and bζ ≡ 1 in a neighborhood of B(0, 1),
• ϕ := ζ1/2− ζ, so that bϕ = bζ(·/2) − bζ and supp bϕ ⊂ B(0, 4) \ B(0, 1).
The (inhomogeneous) Littlewood-Paley decomposition of a temperate dis-tribution f ∈ S′(Rn) is then given by
(9) f =X
j≥0
fj, where f0 = f ∗ ζ and fj = f ∗ ϕ21−j for j ≥ 1.
See for instance [4, Section VI.4.1].
The Littlewood-Paley decomposition can be used to characterize the space Bp,qs [7, Section 2.3.2, Proposition 1, p. 46], and this is the definition we adopt here: (10) Bp,qs = f ∈ L p; |f |q Bs p,q := X j≥0 2sjqkfjkqLp < ∞ . The norm on Bs p,q is defined by (11) kf kqBs p,q = kf k q Lp+ |f |qBs p,q.
Different choices of ζ yield equivalent norms [8, Section 2.3]. See also [8, Chapter 3] for other equivalent characterizations of Bp,qs .
2.2. Schur’s criterion. We will also make use of the following Schur-type estimate for kernel operators; see e.g. [2, Appendix I].
Lemma 2.1. Let (X, µ) and (Y, ν) be two (σ-finite) measure spaces, let
1 ≤ p ≤ ∞, and κ : X × Y → C a measurable kernel. If the quantities M1:= esssupx
ˆ
|κ(x, y)| dν(y) and M2 := esssupy
ˆ
|κ(x, y)| dµ(x),
are finite, then the formula
T u(x) = ˆ
κ(x, y)u(y) dν(y)
defines a bounded linear operator from Lp(Y ) to Lp(X), with norm
kT k ≤ M1/p
′
Here p′= p/(p − 1) is the conjugate exponent of p.
3. Proof of Theorem 1.1
The proof of Theorem1.1 relies on the following ingredient. Lemma 3.1. Let ρ ∈ L1, and let ψ ∈ L1 satisfy ´ ψ = 0. Then
lim
ε→0kρ ∗ ψεkL
1 = 0.
More generally, for ρ and ψ as above we have the following uniform esti-mate:
(12) lim
ε→01/2≤δ≤1sup kρδ∗ ψεkL
1 = 0.
Proof of Theorem 1.1. We are going to prove a discrete version of 4. We start from the inequalities
(13) X j≥0 2sjq ˆ 1 1/2 kf − f ∗ ρ2−jεkqLpdε ≤ ˆ 1 0 kf − f ∗ ρεkqLp dε εsq+1 ≤2sq+1X j≥0 2sjq ˆ 1 1/2 kf − f ∗ ρ2−jεkqLpdε.
In view of 13, it suffices to establish the estimate (14) kf kqBs p,q ≤ C(s, p, q) kfkq Lp+ X j≥0 2sjqkf − f ∗ ρ2−jεkqLp ,
uniformly with respect to ε ∈ (1/2, 1). Integrating 14 and using 13, we obtain indeed the desired inequality 4.
To simplify the notation, we will establish14for ε = 1, which amounts to considering eρ = ρε instead of ρ. It will be clear at the end of the proof that
all estimates are indeed uniform with respect to ε ∈ (1/2, 1). We introduce a function ψ ∈ S satisfying the following: (15) ψ ≡ 1 on supp bb ϕ, and bψ(0) = 0.
Recall that ϕ is the function used in the definition of the Littlewood-Paley decomposition 9. Since the support of bϕ is contained in the annulus {1 ≤ |ξ| ≤ 4}, it is indeed possible to choose ψ satisfying 15.
We need to estimate the Bp,qs semi-norm of f , hence the sum X
j≥0
where f =Pjfj is the Littlewood-Paley decomposition9. We introduce an
integer k > 0, to be fixed later, and split the sum into two parts: (16) |f |qBs p,q ≤ X j≤k 2sjqkfjkqLp+ X j>k 2sjqkfjkqLp.
Using the fact that
kfjkLp = kf ∗ ϕ21−jkLp ≤ kf kLpkϕkL1, ∀ j ≥ 1,
and kf0kLp = kf ∗ ζkLp≤ kf kLpkζkL1,
we simply estimate the first sum in the right-hand side of 16 by
(17) X
j≤k
2sjqkfjkqLp .kf kqLp.
We next turn to estimating the second sum. In the remaining part of the proof, we will use the notation
ρj := ρ2−j, ϕj := ϕ2−j, ψj := ψ2−j.
Taking advantage of the fact that ψ ∗ ϕ = ϕ (and thus ψj∗ ϕj = ϕj) we
write, for j > k,
fj+1= (f − f ∗ ρj−k+ f ∗ ρj−k) ∗ ϕj
= (f − f ∗ ρj−k) ∗ ϕj+ f ∗ ρj−k∗ ψj∗ ϕj = (f − f ∗ ρj−k) ∗ ϕj+ fj+1∗ (ρ ∗ ψk)j−k.
We deduce the estimate
(18) kfj+1kLp≤ kϕkL1kf − f ∗ ρj−kkLp+ kρ ∗ ψkkL1kfj+1kLp.
Since bψ(0) = 0, we can apply Lemma 3.1above: it holds (19) kρ ∗ ψkkL1 = kρ ∗ ψ
2−kkL1 → 0, as k → ∞.
Thus for sufficiently large k we may absorb the last term of the right-hand side of 18 into the left-hand side. For such k, we have
(20) kfj+1kLp.kf − f ∗ ρj−kkLp for j ≥ k.
Plugging 20 into16 and recalling17, we obtain (21) kf kqBs p,q .kf k q Lp+ X j≥0 2sjqkf − f ∗ ρ2−jkqLp.
The latter estimate is exactly the desired estimate 14 with ε = 1. The corresponding estimate for 1/2 ≤ ε ≤ 1 is found by replacing ρ with eρ = ρεin
Proof of Lemma 3.1. We introduce a parameter R > 0. Taking advantage
of the fact that ´ ψ = 0, we may write
(22) ρ ∗ ψε(x) = 1 εn ˆ ρ(y) − BRε(x) ρψ x − y ε dy = 1 Rnε2nω n ¨ |z−x|<Rε (ρ(y) − ρ(z))ψ x − y ε dydz, where BRε(x) is the open ball of center x and radius Rε, and ωn is the
Lebesgue measure of the unit ball. We then have (23) ˆ |ρ ∗ ψε(x)| dx ≤ ˆ MR(x) dx + ˆ NR(x) dx, where MR(x) = 1 Rnε2nω n ¨ |z−x|<Rε |y−x|<Rε |ρ(y) − ρ(z)| ψ x − y ε dydz, (24) NR(x) = 1 Rnε2nω n ¨ |z−x|<Rε |y−x|≥Rε (|ρ(y)| + |ρ(z)|) ψ x − y ε dydz. (25)
To estimate ´ MR(x) dx, we perform the change of variable x w =
(x − y)/ε and find ˆ MR(x) dx ≤ 1 Rnεnω n ˆ |w|<R |ψ(w)| dw ¨ |z−y|<2Rε |ρ(y) − ρ(z)| dydz ≤ kψkL1 Rnεnω n ˆ |h|<2Rε kρ(· + h) − ρkL1dh, and thus (26) ˆ MR(x) dx ≤ 2nkψkL1 sup |h|<2Rε kρ(· + h) − ρkL1.
Note that, for any fixed R, the right-hand side of26converges to 0 as ε → 0. We next estimate ´ NR(x) dx. To this end we compute
Plugging 27 and28 into formula25, we obtain (29) ˆ NR(x) dx ≤ 2kρkL1 ˆ |w|≥R |ψ(w)| dw. Combining23,26 and29 we obtain
lim sup ε→0 kρ ∗ ψεkL1 ≤ Ckρk L1 ˆ |w|≥R |ψ(w)| dw,
and complete the proof of the first assertion in Lemma3.1by letting R → ∞. Estimate 12follows from the following calculations:
lim ε→01/2≤δ≤1sup kρδ∗ ψεkL 1 = lim ε→01/2≤δ≤1sup k(ρ ∗ ψε/δ)δkL 1 = lim ε→01/2≤δ≤1sup kρ ∗ ψε/δkL 1 = lim ε→0kρ ∗ ψεkL 1. 4. Proof of Theorem 1.2
Proof of Theorem 1.2. We clearly have “2 =⇒ 1”, and it remains to prove
that “1 =⇒ 2”. For the convenience of the reader, we start by establishing a consequence of property 1, and then we proceed to the proof of the desired implication.
Step 1. A discrete-uniform version of 1.
Assume that property 1 holds. Then we claim that for every ϕ ∈ S we have
(30) sup
1/2≤ε≤1
X
j≥0
2sjkϕ − ϕ ∗ ρ2−jεkL1 ≤ C < ∞.
In order to prove 30, we start from the following fact. We fix a function λ ∈ S such that ´ λ 6= 0. Then every function ψ ∈ S (Rn) may be written
as
(31) ψ =X
k≥0
λkψ∗ λ2−k.
Here (λkψ)k⊂ S is a sequence that decays rapidly as k → ∞, in the following
sense: if ψ belongs to a bounded subset B ⊂ S, then for every M > 0 there exists a constant C such that
(32) kλkψkL1 ≤
C
2M k, ∀ k ≥ 0, ∀ ψ ∈ B;
We now choose an appropriate λ ∈ S. In view of13, if property 1 holds then we may find some ε ∈ [1/2, 1] such that λ := η1/ε satisfies
(35) X k≥0 2skkλ − λ ∗ ρ2−kkL1 = X k≥0 2skkη − η ∗ ρ2−kεkL1 < ∞.
By combining 33-35 we find that, with ε ∈ [1/2, 1] and t := 1/ε ∈ [1, 2], we have X j≥0 2sjkϕ − ϕ ∗ ρ2−jεkL1 = X j≥0 2sjkϕt− ϕt∗ ρ2−jkL1 ≤X j≥0 2sjX k≥0 kλk,t∗ λ2−k− λk,t∗ λ2−k∗ ρ2−jkL1 ≤X j≥0 X k≥0 2sjkλk,tkL1kλ 2−k− λ2−k∗ ρ2−jkL1 ≤CX j≥0 X k>j 2sjkλk,tkL1 +X j≥0 X k≤j 2sjkλk,tkL1kλ − λ ∗ ρ 2k−jkL1 ≤CX j≥0 X k>j 2sj2−(s+1)k +X ℓ≥0 X j≥ℓ 2sjkλj−ℓ,tkL1kλ − λ ∗ ρ 2−ℓkL1 ≤C + CX ℓ≥0 X j≥ℓ 2sj2−(s+1)(j−ℓ)kλ − λ ∗ ρ2−ℓkL1 ≤C + CX ℓ≥0 2sℓkλ − λ ∗ ρ2−ℓkL1 ≤ C,
with constants independent of t, i.e., 30 holds.
Step 2. Proof of “1 =⇒ 2”.
As we proved in the previous step, we may assume that there exists some η ∈ S such that
(36) bη ≡ 1 in B(0, 4),
and such that η satisfies the following uniform and discrete version of 6: (37) Sε:=
X
j≥0
2sjkη − η ∗ ρ2−jεkL1 ≤ C, ∀ ε ∈ [1/2, 1],
with C independent of ε ∈ [1/2, 1].
Let f ∈ Lp. We will establish the estimate
(38) X
j≥0
2sjqkf − f ∗ ρ2−jεkLqp ≤ C (1 + Sε)q |f |qBs
p,q, ∀ ε ∈ [1/2, 1],
In turn, estimate38 is obtained as follows. Set (39) αj,ε:= 2sjkη − η ∗ ρ2−jεkL1, which satisfies
X
j≥0
αj,ε≤ C, ∀ ε ∈ [1/2, 1].
Let f =Pℓ≥0fℓbe the (inhomogeneous) Littlewood-Paley decomposition
of f ∈ Lp, defined in Section2.1. By 36, for every ℓ we have fℓ = fℓ∗ η2−ℓ,
and thus (40) f − f ∗ ρ2−jε= X ℓ≥0 (fℓ− fℓ∗ ρ2−jε) =X ℓ≥j (fℓ− fℓ∗ ρ2−jε) + X ℓ<j (fℓ− fℓ∗ ρ2−jε) =X ℓ≥j (fℓ− fℓ∗ ρ2−jε) + X ℓ<j fℓ∗ (η2−ℓ− η2−ℓ∗ ρ2−jε) =X ℓ≥j (fℓ− fℓ∗ ρ2−jε) + X ℓ<j fℓ∗ (η − η ∗ ρ2ℓ−jε)2−ℓ.
Using40, we find that (41) kf − f ∗ ρ2−jεkLp. X ℓ≥j kfℓkLp+ X ℓ<j 2−s(j−ℓ)αj−ℓ,εkfℓkLp, i.e., (42) 2sjkf −f ∗ρ2−jεkLp . X ℓ h 2s(j−ℓ)1{ℓ≥j}(ℓ) + αj−ℓ,ε1{ℓ<j}(ℓ) i 2sℓkfℓkLp.
We obtain38by combining39with42and with Schur’s criterion (Lemma2.1) applied to:
X = Y = Z+, µ = ν = the counting measure on Z+,
and k(j, ℓ) = 2s(j−ℓ)1{ℓ≥j}(ℓ) + αj−ℓ,ε1{ℓ<j}(ℓ), ∀ j, ℓ ∈ Z+.
We continue with another characterization of the kernels ρ satisfying the equivalent properties 1 and 2 in Theorem 1.2. For simplicity, the main results of our article were stated for inhomogeneous Besov spaces. It turns out that the homogeneous version of our next result is easier to understand than the inhomogeneous one, so that we start by presenting (without proof) the homogeneous cousin of Theorem4.1 below.
In order to avoid subtle issues concerning the realization of homogeneous Besov spaces as spaces of distributions, we consider only temperate distri-butions f such that
(43) f is compactly supported in Rb n\ {0}.
Any such f is smooth, and we have f =Pj∈Zfj in S′, where (in the spirit
of 9) fj = f ∗ ϕ21−j, ∀ j ∈ Z. For f satisfying 43, we set
with the obvious modification when q = ∞. Let us note that, the series P
j∈Zfj containing only a finite number of non zero terms, we actually have
˙
Bp,qs = {f ∈ Lp(Rn); f satisfies 43},
but that the norm we consider is not equivalent to the Lp norm.
As in the inhomogeneous case considered in this article, we may try to characterize the L1 kernels ρ such that
(44) |f |q˙ Bs p,q ∼ ˆ ∞ 0 1 εsq+1kf − f ∗ ρεk q
Lpdε, for every f satisfying 43.
The homogeneous counterpart of Theorem1.2consists of the following equiv-alence: for a fixed s (not necessarily positive) 44 holds if and only if for a function ϕ as in the Littlewood-Paley decomposition we have
(45)
ˆ ∞
0
1
εs+1kϕ − ϕ ∗ ρεkL1dε < ∞.
Necessity of45 comes from the fact that 44 holds with p = q = 1.
Let us now examine what is required in order to have44when p = q = ∞. If 44holds and if |f |B˙s
∞,∞ < ∞, then the distribution
f − f ∗ ρε= (δ − ρ)ε∗ f
is well-defined (as the convolution of a finite measure with a smooth bounded function).2 Moreover, kf −f ∗ρεkL∞ is controlled by the norm |f |˙
Bs
∞,∞ (since
44 holds). A moment thought shows that in particular δ − ρ is an element of the dual of ˙Bs
∞,∞. Remarkably, this necessary condition is also sufficient,
and is equivalent to the property45.
Theorem4.1is the inhomogeneous counterpart of the above fact. In order to state this result, it is convenient to define ad hoc norm and function space. Fix ζ, ϕ as in the Littlewood-Paley decomposition 9. In order to simplify the proof of Theorem 4.1, we make the (unessential) assumption that (46) ϕ is even.
Our appropriate function space is defined starting from the identity (47) f = (f − f ∗ ζ) + X j≤−1 f ∗ ϕ2−j := X j≤0 fj♯, ∀ f ∈ S′ satisfying 43.
We define the appropriate norm (48) [f ]qXs
p,q =
X
j≤0
2sjqkfj♯kqLp,
with the corresponding modification when q = ∞. Let Xs
p,q be the space of
temperate distributions satisfying 43and such that [f ]Xs
p,q < ∞. 3
2Here, δ stands for the Dirac mass at the origin.
Theorem 4.1. Let s > 0. Then property 6 is equivalent to
(49) δ − ρ ∈ X∞,∞s ∗.
Proof.
“6 =⇒ 49”. Let ϕ be as in the Littlewood-Paley decomposition and let ψ be as in 15. We may assume that ψ is even. If f ∈ S′ and ε > 0 are such that f ∗ ϕε∈ L∞, then we have
(δ − ρ)(f ∗ ϕε) = (δ − ρ)(f ∗ ϕε∗ ψε) = [(δ − ρ) ∗ ψε](f ∗ ϕε)
= ˆ
[(δ − ρ) ∗ ψε(x)] [f ∗ ϕε(x)] dx.
In particular, if j < 0 and f ∈ X∞,∞s , then
(50) (δ − ρ)(fj♯) = ˆ (δ − ρ) ∗ ψ2−j(x) fj♯(x) dx ≤ k(δ−ρ)∗ψ2−jkL1kf♯ jkL∞.
On the other hand, for j = 0 we have f0♯ ∈ C∞∩ L∞ (in view of 43 and of the definition of X∞,∞s ) and thus
(51) (δ − ρ)(f0♯) ≤ (1 + kρkL1)kf♯
0kL∞.
We next note that30 (applied to ψ instead of ϕ), which is a consequence of 6, implies that (52) X j<0 2−sjk(δ − ρ) ∗ ψ2−jkL1 = X j<0 2−sjkψ2−j− ρ ∗ ψ2−jkL1 =X j<0 2−sjkψ − ψ ∗ ρ2jkL1 =X k>0 2skkψ − ψ ∗ ρ2−kkL1 < ∞. By combining 50–52, we obtain |(δ − ρ)(f )| .X j≤0 (δ − ρ)(fj♯) . kf0♯kL∞+ X j<0 k(δ − ρ) ∗ ψ2−jkL1kf♯ jkL∞ ≤kf0♯kL∞+ sup j<0 2sjkfj♯kL∞ X j<0 2−sjk(δ − ρ) ∗ ψ2−jkL1 .kf kXs ∞,∞,
“49=⇒6”. We start by noting that an equivalent formulation of 49 is
(53)
f =X
j∈J
fj♯, with fj♯ as in47 and J ⊂ Z− finite
=⇒ (δ − ρ) X j∈J fj♯ .supj∈J 2sjkfj♯kL∞.
Step 1 in the proof of Theorem 1.2 implies that, if we find some λ ∈ S such that ´ λ 6= 0 and
(54) X
j≥0
2sjkλ − λ ∗ ρ2−jkL1 < ∞,
then 6holds.
Let ζ, ϕ be as in the Littlewood-Paley decomposition. We will prove that
54 holds with λ = ζ. Set
αj := kϕ2j−ϕ2j∗ρkL1 = k(ϕ−ϕ∗ρ2−j)2jkL1 = k(ϕ−ϕ∗ρ2−j)kL1, ∀ j > 0.
We divide the proof of 54into two steps.
Step 1. It suffices to prove the key estimate
(55) X
j>0
2sjαj < ∞.
Granted55, we prove54 for λ = ζ. Indeed, using the fact that lim
M →∞kζM − ζM ∗ ρkL
1 = lim
M →∞kζ − ζ ∗ ρ1/MkL
1 = 0,
we find that, in L1, we have
(56) ℓ→∞lim ℓ X j=k+1 (ϕ2j− ϕ2j∗ ρ) = lim ℓ→∞[(ζ2k− ζ2k∗ ρ) − (ζ2ℓ− ζ2ℓ∗ ρ)] = ζ2k − ζ2k∗ ρ. By 56, we have (57) kζ2k − ζ2k∗ ρkL1 ≤ X j≥k+1 αj.
By combining 55with 57, we obtain X k≥0 2skkζ − ζ ∗ ρ2−kkL1 = X k≥0 2skkζ2k− ζ2k∗ ρkL1 ≤ X k≥0 X j≥k+1 2skαj .X j>0 2sjαj < ∞,
and thus 54 holds.
For ℓ < 0, let ψℓ ∈ C∞
c (Rn) be such that |ψℓ| ≤ 1 and
(58) ˆ [(δ − ρ) ∗ ϕ2−ℓ]ψℓ≥ 1 2k(δ − ρ) ∗ ϕ2−ℓkL1 = 1 2α−ℓ. Let J ⊂ Z∗
− be a fixed arbitrary finite set, and set
f :=X ℓ∈J 2−sℓψℓ∗ ϕ2−ℓ. By 58, we have (using46) (59) X ℓ∈J 2−sℓα−ℓ . X ℓ∈J 2−sℓ ˆ [(δ − ρ) ∗ ϕ2−ℓ] ψℓ = (δ−ρ) X ℓ∈J 2−sℓψℓ∗ ϕ2−ℓ ! . By 53 and 59, we have (60) X ℓ∈J 2−sℓα−ℓ . sup j∈M 2sjkfj♯kL∞,
where M ⊂ Z− is finite and such that fj♯ = 0 when j 6∈ M .4
We next note that, when j, ℓ < 0, we have (61) ϕ2−ℓ∗ ϕ2−j = 0 when |j − ℓ| > 1. By 61, when j < 0 we have (62) fj♯ =X ℓ∈J 2−sℓψℓ∗ ϕ2−ℓ ♯ j = X ℓ∈J 2−sℓψℓ∗ ϕ2−ℓ∗ ϕ2−j = X ℓ∈J |ℓ−j|≤1 2−sℓψℓ∗ ϕ2−ℓ∗ ϕ2−j, and thus (63) kfj♯kL∞ . X ℓ∈J |ℓ−j|≤1 2−sℓkψℓkL∞ .2−sj. By 60 and 63, we have (64) X ℓ∈J 2−sℓα−ℓ ≤ C < ∞, with C independent of J.
We obtain55 by taking, in 64, the supremum over J.
5. Further results
This section is devoted to the proofs of Propositions 1,2 and 3.
5.1. Proof of Proposition 1. Proposition1 is a direct consequence of the following more general result.
Proposition 4. Let ρ ∈ L1 satisfy ´ ρ = 1 and let η ∈ S be such that
´ η 6= 0. Assume that ρ has finite moments of any order: ˆ
|y|k|ρ(y)| dy < ∞ for all k ∈ N. Then (65) ˆ 1 0 1 εs+1kη − η ∗ ρεkL1dε < ∞ if and only if s < k0,
where k0 ∈ N∗∪ {∞} is the smallest non-zero moment of ρ:
k0 = min k ≥ 1 : ˆ y⊗kρ(y) dy 6= 0 .
Here y⊗k denotes the k-th order tensor (yj1· · · yjk)1≤j1,...,jk≤n.
Note that Proposition4 implies indeed Proposition1since for a bounded set A of positive measure the second moment ´
Ay⊗2dy is always non zero.
We now turn to the
Proof of Proposition 4. We first treat the case of a finite k0. Since it holds
η(x) − η ∗ ρε(x) =
ˆ
(η(x) − η(x − εy))ρ(y) dy, we find, applying Taylor’s formula,
η(x) − η ∗ ρε(x) =(−1) k0+1 k0! εk0 X 1≤j1,...,jk0≤n αj1,...,jk0∂j1· · · ∂jk0η(x) + ε k0+1 Rε(x), where (66) αj1,...,jk := ˆ yj1· · · yjkρ(y) dy, and kRεkL1 ≤ kDk0+1ηk L1 (k0+ 1)! ˆ
|y|k0+1|ρ(y)| dy.
Therefore it holds (67) kη−η∗ρεkL1 = 1 k0! εk0 X 1≤j1,...,jk0≤n αj1,...,jk0∂j1· · · ∂jk0η L1+O(ε k0+1 ), as ε → 0.
Indeed, assume that c = 0. Then we have X
1≤j1,...,jk0≤n
αj1,...,jk0ξj1· · · ξjk0η(ξ) = 0 ∀ ξ ∈ Rˆ
n.
Since ˆη(0) 6= 0 we deduce that X
1≤j1,...,jk0≤n
αj1,...,jk0ξj1· · · ξjk0 = 0
for all sufficiently small ξ, and thus by homogeneity for every ξ. This is absurd since, by assumption, at least one of the coefficients αj1,...,jk0 is non
zero.
Therefore c 6= 0 and the Taylor expansion67 provides the equivalent kη − η ∗ ρεkL1 ∼
c k0!
εk0
as ε → 0, which readily implies65. This concludes the proof of Proposition4
when k0 is finite.
When k0 = ∞, the Taylor expansion shows that
kη − η ∗ ρεkL1 = O(εk) for all k ∈ N,
so that it holds indeed ˆ 1
0
1
εs+1kη − η ∗ ρεkL1dε < ∞
for every s > 0.
5.2. Proof of Proposition 2. We fix ρ ∈ L1 with ´ ρ = 1 and 0 < s < 1, and assume that ρ satisfies the moment condition8:
ˆ
|y|s|ρ(y)| dy < ∞.
We consider an arbitrary test function η ∈ S and are going to show that condition 6 is satisfied (so that, by Theorem 1.2, the norm equivalence7 is valid). To this end we compute
ˆ ∞ 0 kη − η ∗ ρεkL1 dε εs+1 ≤ ˆ ∞ 0 ˆ kη − η(· − εy)kL1|ρ(y)| dy dε εs+1 = ˆ |y|sρ(y) ˆ ∞ 0 kη − η(· − εy)kL1 |εy|s dε ε dy = ˆ |y|sρ(y) ˆ ∞ 0 kη − η(· − δ y |y|)kL1 dδ δs+1dy.
On the other hand, for every ω ∈ Sn−1 we have the estimate
and therefore we conclude that ˆ ∞ 0 kη − η ∗ ρεkL1 dε εs+1 ≤ C(η) ˆ |y|s|ρ(y)| dy < ∞,
which finishes the proof of Proposition 2.
5.3. Proof of Proposition 3. Let s > 0 and let ρ ∈ L1 satisfy ´ ρ = 1 and ρ ≥ 0. We assume that the norm equivalence 7 is valid. Then by Theorem1.2 (and Step 1 in its proof), it holds
ˆ 1
0
kη − η ∗ ρεkL1
dε εs+1 < ∞
for every η ∈ S. We fix such a function η ≥ 0, η 6≡ 0, with support in the unit ball:
η(x) = 0 for |x| ≥ 1. We are going to show that (68) ˆ 1 0 kη−η∗ρεkL1 dε εs+1 ≥ ckηkL1 ˆ
|y|sρ(y) dy−C(kηkL1+kηkL∞kρk
L1),
for some constants c = c(s), C = C(s) > 0. Obviously68implies the conclu-sion of Proposition 3: the function ρ satisfies the finite moment condition
ˆ
|y|sρ(y) dy < ∞.
We now turn to the proof of68. Note that ˆ ∞ 1 1 εs+1kη − η ∗ ρεkL1dε ≤ ˆ ∞ 1 dε εs+1(kηkL1 + kηkL∞kρkL1).
Hence it suffices to show that ˆ ∞
0
1
εs+1kη − η ∗ ρεkL1dε ≥ ckηkL1
ˆ
|y|sρ(y) dy.
Since η(x) = 0 for |x| ≥ 1, and since η and ρ are non negative, it holds kη − η ∗ ρεkL1 ≥ ¨ |x|≥1 η(x − εy)ρ(y) dy = ¨ |z+εy|≥1 η(z)ρ(y) dydz. Thus we obtain (69) ˆ ∞ 0 1 εs+1kη − η ∗ ρεkL1dε ≥ ˚ |z+εy|≥1 η(z)ρ(y) εs+1 dydzdε = ˚ |z+δy/|y||≥1 η(z)ρ(y)|y| s δs+1 dydzdδ.
Note that it holds
[|δ| ≥ 2 and |z| < 1] =⇒ |z + δy/|y|| ≥ 1.
Therefore, the domain of integration in the last integral in 69 contains the set
We find that ˆ ∞ 0 1 εs+1kη − η ∗ ρεkL1dε ≥ kηkL1 ˆ ∞ 2 dδ δs+1 ˆ ρ(y)|y|sdy,
which completes the proof of 68.
Acknowledgments
Part of this work was carried out while XL was visiting McMaster Uni-versity. He thanks the Mathematics and Statistics department of McMaster University, and in particular S. Alama and L. Bronsard, for their hospitality. PM was partially supported by the ANR project “Harmonic Analysis at its Boundaries”, ANR-12-BS01-0013-03. Both authors were partially supported by the Laboratoire d’excellence MILYON, ANR-10-LABX-0070.
References
[1] G. Bourdaud, Ondelettes et espaces de Besov, Rev. Mat. Iberoamericana, 11 (1995), 477–512.
[2] L. Grafakos, Classical Fourier analysis, vol. 249 of Graduate Texts in Mathematics, 2nd edition, Springer, New York, 2008.
[3] E. Stein, An H1
function with nonsummable Fourier expansion, in Harmonic analysis (Cortona, 1982), vol. 992 of Lecture Notes in Math., Springer, Berlin, 1983, 193–200. [4] E. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory inte-grals, vol. 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 1993, With the assistance of Timothy S. Murphy, Monographs in Harmonic Anal-ysis, III.
[5] E. Stein, M. Taibleson and G. Weiss, Weak type estimates for maximal operators on certain Hpclasses, in Proceedings of the Seminar on Harmonic Analysis (Pisa, 1980),
vol. suppl. 1, 1981, 81–97.
[6] H. Triebel, Interpolation theory, function spaces, differential operators, vol. 18 of North-Holland Mathematical Library, North-North-Holland Publishing Co., Amsterdam, 1978. [7] H. Triebel, Theory of function spaces, vol. 78 of Monographs in Mathematics,
Birkh¨auser Verlag, Basel, 1983.
[8] H. Triebel, Theory of function spaces. II, vol. 84 of Monographs in Mathematics, Birkh¨auser Verlag, Basel, 1992.
